Preheat oven to 350°F. Grease inside of a 10-inch springform pan with about 1 tsp. butter, and then set it aside.
Make crust: Combine graham cracker crumbs and 1/4 cup sugar in a bowl, stirring with a fork until blended. Add butter and stir until incorporated. Press crumb mixture into bottom of prepared pan. Bake for 10 minutes; let cool on a wire rack.
Make filling: Increase oven temperature to 450°F. In bowl of an electric mixer, beat cream cheese and 1 cup sugar at high speed until fluffy, about 5 minutes. Add eggs all at once and mix at medium speed until thoroughly blended. Add sour cream and vanilla extract and mix until blended. Scrape batter into pan with crust and bake at 450°F for 15 minutes. Do not open oven door. Reduce oven temperature to 200°F and bake for 1 hour. Turn oven off; open oven door partially. Let cake cool inside oven for 1 hour.
Chill cheesecake until cold, about 3 hours. Top with blackberries, blueberries and raspberries when ready to serve.